Doppler shift and spread estimation method and apparatus

ABSTRACT

For a noise resistant estimation of a Doppler spread for a radio signal transmission channel in a mobile communications environment, the Doppler spread is estimated on the basis of an autocorrelation function for predefined signal portions of a received radio signal, such as pilot bits of signals transmitted via a control channel in a mobile telephone system. The received radio signal and in particular the predefined signal portions are demodulated to obtain discrete autocorrelation coefficients for the predefined signal portions. First and second derivatives of the autocorrelation function are estimated on the basis of at least two of the autocorrelation coefficients. In dependence of used autocorrelation coefficients, influence of noise included in the received radio signal can be at least reduced with respect to an estimation of the Doppler spread. Using the autocorrelation coefficients being indicative of the autocorrelation function and of the estimated first and second derivatives thereof, the Doppler spread can be estimated.

FIELD OF THE INVENTION

The present invention relates to radio signal processing in mobilecommunications environments. In particular, the present inventionrelates to a method and apparatus for Doppler shift and spreadestimations based on a correlation of demodulated pilot data of a radiosignal transmitted in a mobile communications environment.

BACKGROUND OF THE INVENTION

Transmission of radio signals, e.g. employed in a mobile communicationsenvironment such as GSM and UTMS systems, are subjected to reflections,dispersions and the like such that a plurality of sub-radio signals aretransmitted to a receiver from different directions. In dependence ofthe phases, the received sub-signals amplify or reduce each other. Thus,the resulting received signal is a function varying with the location ofthe receiver and, in case of a moving receiver, varying in time.Additionally, a movement of a transmitter results in a frequency shiftof received signals due to the Doppler effect.

In order to compensate frequency shifts and spreads of received radiosignals, the so-called Doppler shift and Doppler spread are used toestimate a frequency shift and frequency spread. The Doppler shift isindicative of a frequency offset which characterizes the difference inmean frequency between transmitted and received signals. The Dopplerspread is indicative of a frequency spread which characterizes in amultipath propagation environment how fast the transmission channel usedfor radio signals is fading.

The Doppler shift which is also referred to as Doppler frequency f_(d)can be computed by:

$\begin{matrix}{f_{d} = {f_{c} \cdot \frac{v}{c}}} & (1)\end{matrix}$wherein f_(c) is the carrier frequency, v is the speed of a movingreceiver (e.g. mobile user equipment such as a mobile telephone) and cis the speed of light.

As shown in FIG. 1, the two channel parameters, frequency offset andDoppler shift, characterize a Doppler spectrum of a wireless radiosignal communications channel.

A mobile radio signal communications channel can be characterized by asequence of complex channel coefficients varying in time:c (k)=Re{c (k)}+jIm{c (k)}  (2)

The channel coefficient sequence can be modeled as time discretestationary random process wherein an autocorrelation sequence r_(cc)(n)is defined as:

$\begin{matrix}{{r_{cc}(n)} = {\lim\limits_{Karrow\infty}{\frac{1}{{2K} + 1}\;{\sum\limits_{k = {- K}}^{K}{{{\underset{\_}{c}}^{*}(k)}\;{c( {k + n} )}}}}}} & (3)\end{matrix}$

Transforming the autocorrelation sequence r_(cc)(n) in the frequencydomain, the respective power density spectrum φ_(cc)(f) is given by:

$\begin{matrix}{{\phi_{cc}(f)} = {\sum\limits_{n = {- \infty}}^{\infty}{{\varphi_{cc}(n)}\;{\mathbb{e}}^{{- 2}\;\pi\;{fTn}}}}} & (4)\end{matrix}$which describes the spectral distribution of an unmodulated signaltransmitted over a mobile radio signal communications channel c. Thispower density function is also called Doppler power density spectrum theshape thereof being dependent of the environment wherein a moving partytransmits or receives a radio signal and of its speed of movement.

In order to predict the shape of a Doppler power density spectrum, shortDoppler spectrum, different modelling methods are employed. For example,for modelling a Doppler spectrum it is assumed that a radio signal fromor to a mobile entity by means of a radio signal channel having only onedirect line of sight to the mobile entity. For this easiest case, theDoppler spectrum can be denoted by:φ_(cc) ^(d)(f)=δ(f−f ₀)  (5)wherein δ(f) denotes a dirac pulse. Here, the signal received by amobile entity is simply shifted in frequency by a frequency offset.

A more complicated model assumes that radio signals are received ortransmitted by a mobile entity by means of a channel for which no directpath is existing to the mobile entity. Thus, the resulting receivedsignal comprises a superposition of signals received via indirect paths.In order to reduce the complexity of the calculation of a Dopplerspectrum for the latter case, especially in case of a mobile radiosignal communications environment, it is generally assumed that thepropagation of signals takes place in a horizontal plane, that the angleof incidence is equally distributed in the range between 0 and 2π forall received signals and that the signal strengths for all signals viaindirect paths are equally distributed. This assumption leads to theso-called Jake's spectrum which is given by:

$\begin{matrix}{{\phi_{cc}^{J}(f)} = \{ \begin{matrix}{\frac{2\;\sigma_{0}^{2}}{\pi\; f_{\max}\sqrt{1 - ( {f/f_{\max}} )^{2}}},} & {{f} \leq f_{\max}} \\{0,} & {{f} > f_{\max}}\end{matrix} } & (6)\end{matrix}$

On the basis of a Doppler spectrum according to equation (6), a Dopplerfrequency estimation method was proposed wherein, after a frequencyoffset compensation, an autocorrelation function for the Dopplerspectrum is calculated, the autocorrelation function is expressed by azero-order Bessel function of the first kind and the first-zero crossingof the autocorrelation function is determined. Further, this approach isperformed on the basis of the slot rate used for the radio signals,wherein a slot is a duration which consists of fields containinginformation, e.g. bits. In particular, the slot rate based approachcalculates a complex correlation function between so-called pilot bitsor groups which usually serve as training signals. For obtaining anautocorrelation function, the sampling period is required to be muchsmaller than the inverse of the highest frequency expected in theDoppler spectrum. For example, the slot rate in an UMTS system is 1.500Hz. Thus, it is impossible to perform a Doppler spread estimation onslot rate basis for frequencies above 750 Hz in an UMTS system.

Further, a Doppler spectrum according to equation (6) cannot be assumedunder all circumstances depending on the communications environment, forexample in rural areas compared with urban areas. Rather, Dopplerspectra actually found in communications environments can be Gaussiandistributed due to radio signal reflections and dispersions, can bedefined by a dirac pulse due to a radio signal received via a channelhaving only one direct line of sight, can be result from a superpositionof different distributions and combinations thereof.

For example, the above mentioned autocorrelation based approach using aDoppler spectrum according to equation (6) provides for ineffectiveDoppler spread estimation in case a line of direct sight component ofreceived radio signals influences the resulting Doppler spectrum, e.g.by introducing a Doppler spectrum peak.

Object of the Invention

The object of the present invention is to overcome the above mentionedproblems and in particular to provide a solution for a Doppler spreadestimation when the autocorrelation function of a Doppler spreaddeviates from a Bessel function due to the existence of a line of sightcomponent.

Solution According to the Invention

The present invention teaches to use a definition for the Doppler spreadof a received radio signal in the time domain. In particular, a Dopplerspread definition is used wherein the Doppler spread is characterized bya function in-time of an autocorrelation function of a received radiosignal and the first and second derivatives of the autocorrelationfunction.

For an estimation of the Doppler spread of a radio signal on the basisof such a definition, the basic idea underlying the present invention isto estimate the first and second derivatives of the autocorrelationfunction determined for the radio signal for a predefined point of time.

For an determination of the autocorrelation function and its first andsecond derivatives, an autocorrelation sequence is defined for the radiosignal by modeling the signal as a time discrete signal. In particular,the autocorrelation sequence is determined for known signal portions ofthe received radio signal such as training sequence signals or so-calledpilot symbols.

On the basis of discrete autocorrelation coefficients for theautocorrelation sequence, the first and second derivatives of theautocorrelation function are estimated. In order to reduce or eliminatedisturbances of the radio signal due to signal noise, it is contemplatedto employ specific autocorrelation coefficients for the estimation ofthe first and second derivatives.

For estimating the Doppler spread, the values for the autocorrelationfunction obtained from the autocorrelation coefficients and theestimations based thereon are used to evaluate the Doppler spreaddefinition.

With respect to a mobile telephone system, such as a GSM or UTMSenvironment, the basic idea underlying the present invention can bedefined as a calculation or estimation of Doppler spread for radiosignals transmitted according to the respective communications standardson the basis of an autocorrelation function of received and demodulatedtraining or pilot signal of a control channel.

Short Description of the Invention

In a greater detail, the present invention provides a method for Dopplerspread estimation for a radio signal transmission channel in a mobilecommunications environment on the basis of a radio signal transmittedvia the transmission channel. For carrying out the method according tothe invention an autocorrelation function for the radio signal isdetermined and a Doppler spread for the radio signal is defined as afunction in time of the autocorrelation function and its first andsecond derivatives for a point of time being zero. While theautocorrelation function for the point of time being zero is determined,the first and second derivates of the autocorrelation function areestimated for the point of time being zero. In particular, thisestimations are obtained by an averaging of respective portions of theautocorrelation function each thereof including the point of time beingzero. The determined and estimated values for the autocorrelationfunction are used to compute the defined Doppler spread function wherebyan estimated value for the Doppler spread is obtained.

For the determination of the autocorrelation function it is possible tomodel the radio signal as a time discrete signal and to determine anautocorrelation sequence for the time discrete signal representing thetransmitted radio signal.

For the determination and estimation of values of the autocorrelationfunction it is possible to calculate autocorrelation coefficients forthe autocorrelation sequence. Optionally it is contemplated to define acorrelation influence length and to determine autocorrelation sequencecoefficients by means of a recursive function.

For obtaining the recursive function used for the determination ofautocorrelation sequence coefficients, a linear or an exponentialaveraging of the autocorrelation sequence is possible.

An estimation of the first and second derivatives of the autocorrelationfunction can be provided by determining respective slopes of theautocorrelation function wherein the estimation for the secondderivatives of the autocorrelation function which can be considered as arespective slope of the first derivative of the autocorrelation functioncan also be calculated on the basis of slopes of the autocorrelationfunction itself.

In the case of an autocorrelation sequence, slopes of theautocorrelation function can be determined by averaging processes forrespective autocorrelation coefficients. In particular, twoautocorrelation coefficients for points of time, which define a timeinterval including the point of time (t=0) for which the first andsecond derivatives of the autocorrelation function are to be estimated,are chosen. By means of a linear averaging of two autocorrelationcoefficients chosen in this manner the slopes of the autocorrelationfunction are respectively determined. Preferably the determination ofeach slope of the autocorrelation function is performed such that onlytwo values of the autocorrelation function are necessary to estimate thefirst and second derivatives thereof. As a result, the estimation of theDoppler spread for the transmission channel can be performed on thebasis of only two values for the autocorrelation function defined forthe radio signal.

In order to reduce disturbances of the radio signal due to signal noisepossible leading to an estimation error for the Doppler spread, it iscontemplated to only use such portions of the autocorrelation functionfor the estimation of the first and second derivatives thereof which arenot affected by signal noise.

Especially such a noise resistant estimation of the Doppler spread canbe accomplished if values for the autocorrelation function are employedwhich do not include autocorrelation function values for the point oftime being zero. Further, it is possible to use single, discreteautocorrelation coefficients forming parts of the autocorrelationsequence to estimate the first and second derivatives of theautocorrelation function. In particular, such a procedure is preferredin the case of modeling the radio signal as a time discrete signal.

Using discrete autocorrelation coefficients for the estimation of thefirst and second derivatives of the autocorrelation function, only twoautocorrelation coefficients can be utilized wherein it is possible todetermine at least one of the used autocorrelation coefficients independence of the correlation influence length defined for theautocorrelation function of the autocorrelation sequence, respectively.

Further optimization for the estimation of the Doppler spread can beobtained by evaluating the signal to noise ratio expected for theestimated second derivative of the autocorrelation function. In view ofa threshold value defined for a signal to noise ratio, the signal tonoise ratio for the estimated second derivative of the autocorrelationfunction is determined for different autocorrelation coefficients of theautocorrelation sequence. In the case no autocorrelation coefficientsleads to a signal to noise ratio for the estimated second derivative ofthe autocorrelation function below the predefined signal to noise ratiothreshold value, the Doppler spread is estimated to be zero. The sameestimation for the Doppler spread can result if further constraints aredefined in dependence of the autocorrelation coefficients with respectto the estimated second derivative of the autocorrelation function. Anestimation of the Doppler spread as described above can be performed if,in particular, the signal to noise ratio for the estimated secondderivative of the autocorrelation function exceeds the defined signal tonoise ratio threshold value for a specific autocorrelation coefficient.Then, this autocorrelation coefficient is used for the estimation of thesecond derivative of the autocorrelation function.

In dependence of the manner the radio signal is transmitted via thetransmission channel, it is possible, after having received the radiosignal, to demodulate the received radio signal in particular in view ofa predefined signal sequence included in the radio signal. Here, theautocorrelation function is determined as an autocorrelation functionfor the demodulated radio signal.

In a similar manner as described above, it is possible to define anautocorrelation sequence for the demodulated radio signal.

Especially in the case the radio signal includes, beside the predefinedsignal sequence, further signal portions it is contemplated to definethe autocorrelation sequence and, in particular, the autocorrelationsequence for demodulated radio signal portions being indicative of thepredefined signal sequence. This can be accomplished by means of arecursive function defining a relation between autocorrelationcoefficients of the autocorrelation sequence. Such a recursivedetermination of autocorrelation coefficients can include averagingprocesses for the autocorrelation sequence, in particular in view of thepercentage of the predefined signal sequence portion in the demodulatedradio signal.

As a preferred embodiment, the present invention is utilized in a mobiletelephone environment such as a GSM or UTMS system. Here, thetransmission channel for which the Doppler spread is to be estimated canbe a control channel such as a DPCCH channel. According to the standardsof such mobile telephone environments, the radio signal used for aDoppler spread estimation comprises at least one frame having subsequentslots each thereof including a number of predefined pilot symbols. Inthis case, the predefined pilot symbols represent the above predefinedsignal sequence. By demodulating the radio signal it is possible toobtain the demodulated pilot symbols per slot for which autocorrelationcoefficients can be determined. On the basis of the autocorrelationcoefficients obtained for the demodulated pilot symbols, the first andsecond derivatives of the autocorrelation function can be estimatedwhich are, in turn, used to estimate the Doppler spread.

Further, the present invention provides a computer program product forcarrying out the above described methods and a receiver for a mobilecommunications environment being adapted to incorporate the abovedescribed methods.

SHORT DESCRIPTION OF THE FIGURES

In the following description of the present invention it will bereferred to the enclosed figures wherein:

FIG. 1 is a diagram showing a exemplary graph of a Doppler spectrum,

FIG. 2 is a diagram illustrating the relative error of an estimation fora second derivative of an auto correlation function for a Dopplerspectrum according to the invention due to additive white noise,

FIG. 3 is a diagram illustrating the relative error for a furtherestimation for a second derivative of an autocorrelation function for aDoppler spectrum according to the present invention due to lowerspectral sampling resolution caused by use of larger correlationcoefficients,

FIG. 4 is a diagram illustrating a sum of relative errors forestimations for a second derivative of an autocorrelation function for aDoppler spectrum according to the present invention due to additivewhite noise and due to a linear proximation error,

FIG. 5 is a schematic illustration of a frame structure for a DPPCH,

FIG. 6 is a schematic illustration of a demodulated radio signal,

FIG. 7 is a schematic illustration of a calculation of anautocorrelation function for a Doppler spectrum according to the presentinvention,

FIG. 8 is a schematic illustration of a calculation of anautocorrelation coefficient for the demodulated signal illustrated inFIG. 6,

FIG. 9 is a schematic illustration of a calculation of a furtherautocorrelation coefficient for the signal illustrated in FIG. 6,

FIG. 10 illustrates a simulation environment for a Doppler spreadestimation according to the present invention,

FIG. 11 shows diagrams of results obtain from the simulation environmentshown in FIG. 10 for an indirect received radio signal,

FIG. 12 shows diagrams of results obtained from the simulationenvironment shown in FIG. 10 for a direct received radio signal, and

FIG. 13 shows diagrams of results obtained from the simulationenvironment shown in FIG. 10 for received superposed direct and indirectsignals.

DESCRIPTION OF A PREFERRED EMBODIMENT

The following description presents a time domain based algorithm for anestimation of a frequency offset and a Doppler spread for a radio signalof a mobile communications environment. The estimation will be based onthe measurement of an autocorrelation function of channel coefficients.The method presented here is designed to be independent from the actualshape of a power spectrum density of a radio signal channel.

In the description, the following symbol definitions will be employed:

-   -   f_(max) highest frequency appearing in a non shifted Doppler        spectrum    -   c(k) complex coefficient of a time variant channel pulse        response    -   r_(cc)(n) time discrete autocorrelation sequence of a signal c    -   φ_(cc)(f) power density spectrum of a signal c    -   T sampling period    -   δ(f) dirac pulse in frequency domain    -   f₀ frequency offset    -   σ₀ variance    -   B⁽¹⁾ Doppler shift    -   B⁽²⁾ Doppler spread    -   φ_(cc)(t) time continuous autocorrelation function of a signal c    -   {dot over (φ)}_(cc)(t) first derivative in respect of time of        time continuous autocorrelation function of a signal c    -   {umlaut over (φ)}_(cc)(t) second derivative in respect of time        of a time continuous autocorrelation function of a signal c    -   L correlation influence length    -   {dot over ({tilde over (φ)}_(cc)(t) estimation for {dot over        (φ)}_(cc)(0)    -   {umlaut over ({tilde over (φ)}_(cc)(t) estimation for {umlaut        over (φ)}_(cc)(0)    -   {tilde over (B)}⁽¹⁾ estimation for Doppler shift B⁽¹⁾    -   {tilde over (B)}⁽²⁾ estimation for Doppler spread B⁽²⁾    -   b(f,t) spectral signal component of a received signal    -   n(k) complex additive white noise    -   N signal power of additive white noise n(κ)    -   κ index for autocorrelation coefficient (lag)    -   φ_(cc) ^((κ))(0) estimation for φ_(cc)(0) based on        autocorrelation coefficient κ    -   {dot over ({tilde over (φ)}_(cc) ^((κ))(0) estimation for {dot        over (φ)}_(cc)(0) based on autocorrelation coefficient κ    -   {umlaut over ({tilde over (φ)}_(cc) ^((κ))(0) estimation for        {umlaut over (φ)}_(cc)(0) based on autocorrelation coefficient κ    -   {tilde over (B)}_(κ) ⁽¹⁾ estimation for B⁽¹⁾ based on        autocorrelation coefficient κ        {tilde over (B)}_(κ) ⁽²⁾ estimation for B⁽²⁾ based on        autocorrelation coefficient κ    -   {tilde over ({umlaut over (φ)}_(cc)(0) estimation for {umlaut        over (φ)}_(cc)(0) based on autocorrelation coefficient κ and        disturbed by additive white noise n(κ)    -   ε_(noise) relative error of the estimation for {umlaut over        (φ)}_(cc)(0) due to additive white noise n(κ)    -   ε_(approx) relative error of the estimation for {umlaut over        (φ)}_(cc)(0)    -   n_(p) number of pilot symbols per slot in the DPCCH channel    -   n_(s) total number of symbols per slot in the DPCCH channel    -   y(n) received signal with influence of slot format    -   r B⁽²⁾/f_(max)    -   v velocity of a mobile entity    -   SNR_(lag) signal to noise ratio for the estimation of {umlaut        over (φ)}_(cc)(0) using a particular autocorrelation coefficient    -   SNR_(thr) thereshold value for SNR_(lag) to decide usage of a        particular autocorrelation coefficient    -   c₀ speed of light

$( {{3 \cdot 10^{8}}\mspace{14mu}\frac{m}{s}} )$

-   -   f_(c) carrier frequency (e.g. 2·10⁹ Hz)    -   c_(d)(k) direct signal component of an undisturbed received        signal    -   a_(d) amplitude of a direct signal component of an undisturbed        received signal    -   P_(d) power of a direct signal component of an undisturbed        received signal    -   c_(d)(k) indirect-signal component of an undisturbed received        signal    -   P_(i) power of an indirect signal component of an undisturbed        received signal    -   μ₁(k) real part of c_(i)(k)    -   μ₂(k) imaginary part of c_(i)(k)    -   N_(i) number of sine signals to generate μ_(i)(k)    -   v_(i.n) amplitude of spectral component of μ_(i)(k)    -   f_(i.n) frequency of spectral component of μ_(i)(k)        Definition of Doppler Shift and Spread

A transmission channel for a mobile communications environment can bedescribed by a sequence c(k) of complex time variant channelcoefficience:c (k)=Re{c (k)}+jIm{c (k)}  (7)

The channel coefficience sequence can be modeled as a time discretestationary random process wherein an autocorrelation sequence r_(cc)(n)can be defined as

$\begin{matrix}{{r_{cc}(n)} = {\lim\limits_{Karrow\infty}{\frac{1}{{2K} + 1}\;{\sum\limits_{k = {- K}}^{K}{{{\underset{\_}{c}}^{*}(k)}\;{c( {k + n} )}}}}}} & (8)\end{matrix}$

Transformed into the frequency domain, the resulting power densityspectrum φ_(cc)(f) is given by

$\begin{matrix}{{\phi_{cc}(f)} = {\sum\limits_{n = {- \infty}}^{\infty}{{\varphi_{cc}(n)}\;{\mathbb{e}}^{{- 2}\;\pi\;{fTn}}}}} & (9)\end{matrix}$

The power density function which is also called Doppler power densityspectrum describes the spectral distribution of an unmodulated radiosignal transmitted over a channel c. On the basis of the Dopplerspectrum, two expressions characterizing the Doppler spectrum can bedefined, namely the Doppler shift B⁽¹⁾ and the Doppler spread B⁽²⁾:

$\begin{matrix}{B^{(1)} = \frac{\int_{- \infty}^{\infty}{f\;{\phi_{cc}(f)}\;{\mathbb{d}f}}}{\int_{- \infty}^{\infty}{{\phi_{cc}(f)}\;{\mathbb{d}f}}}} & (10) \\{and} & \; \\{B^{(2)} = \sqrt{\frac{\int_{- \infty}^{\infty}{( {f - B^{(1)}} )^{2}\mspace{11mu}{\phi_{cc}(f)}\;{\mathbb{d}f}}}{\int_{- \infty}^{\infty}{{\phi_{cc}(f)}{\mathbb{d}f}}}}} & (11)\end{matrix}$

The Doppler shift B⁽¹⁾ can be interpreted as a center of gravity of theDoppler spectrum φ_(cc)(f). Since equations (10) and (11) are based onthe Doppler spectrum φ_(cc)(f) and require integration's in thefrequency domain, these equations are difficult to compute. This problemcan be solved by computing the Doppler shift B⁽¹⁾ and the Doppler spreadB⁽²⁾ in the time domain. For that purpose the time continuousautocorrelation function φ_(cc)(t) is calculated as an inverse Fouriertransformation of φ_(cc)(f):

$\begin{matrix}{{\varphi_{cc}(t)} = {\int_{- \infty}^{\infty}{{{\phi_{cc}(f)} \cdot {\mathbb{e}}^{j\; 2\;\pi\;{ft}}}{\mathbb{d}f}}}} & (12)\end{matrix}$

Further, the first and second derivatives of the autocorrelationfunction φ_(cc)(t) with respect of t are calculated by:

$\begin{matrix}{{{{\overset{.}{\varphi}}_{cc}(t)} = {\int_{- \infty}^{\infty}{{j2}\;\pi\; f\;{\phi_{cc}(f)}{\mathbb{e}}^{{j2\pi}\;{ft}}\ {\mathbb{d}f}}}}{and}} & (13) \\{{{\overset{¨}{\varphi}}_{cc}(t)} = {- {\int_{- \infty}^{\infty}{4\pi^{2}f^{2}{\phi_{cc}(f)}\ {\mathbb{e}}^{{j2\pi}\;{ft}}{{\mathbb{d}f}.}}}}} & (13)\end{matrix}$

For an elimination of the integration terms in equations (10) and (11),t is set to 0 in equations (12), (13) and (14). As a result, the Dopplershift B⁽¹⁾ and the Doppler spread B⁽²⁾ are given by:

$\begin{matrix}{{B^{(1)} = {\frac{1}{2\pi\; j}\frac{{\overset{.}{\varphi}}_{cc}(0)}{\varphi_{cc}(0)}}}{and}} & (15) \\{B^{(2)} = {\frac{1}{2\pi}\sqrt{( \frac{{\overset{.}{\varphi}}_{CC}(0)}{\varphi_{CC}(0)} )^{2} - \frac{{\overset{¨}{\varphi}}_{CC}(0)}{\varphi_{CC}(0)}}}} & (16)\end{matrix}$

Thus, the Doppler spread B⁽¹⁾ and the Doppler shift B⁽²⁾ can be easilycomputed in case the autocorrelation function of the respective channelcoefficient sequence is known, derived twice and evaluated for t=0. Ithas to be noted that equations (15) and (16) are equivalent to equations(10) and (11) since no type of Doppler spectrum has been assumed.

Recursive Calculation of the Autocorrelation Sequence

On the basis of equation (7) defining the autocorrelation sequence for atime discrete signal, the n^(th) coefficient for a complex signal s at atime m can be estimated using the following recursive expression:

$\begin{matrix}{{r_{SS}( {n,m} )} = {\lim\limits_{K->\infty}{\frac{1}{{2K} + 1}{\sum\limits_{k = {- K}}^{K}{\underset{\_}{\; s}*( {k + m} ){\underset{\_}{s}( {k + n + m} )}}}}}} & (17)\end{matrix}$

Assuming the signal s is a causal signal s(t) with s(t)=0 for t>m at agiven time m and further assuming a correlation influence length for thecorrelation to be calculated is given by L, an average value of theproduct s*(k)s(k+n) can be calculated for a period L. Thus, thefollowing equation is obtained:

$\begin{matrix}{{r_{SS}( {n,m} )} = {\frac{1}{L - n}{\sum\limits_{k = {1 - L}}^{- n}\;{\underset{\_}{s}*( {k + m} ){\underset{\_}{s}( {k + n + m} )}}}}} & (18)\end{matrix}$

For a time m+1, the n^(th) coefficient r_(ss) is given by:

$\begin{matrix}\begin{matrix}{{r_{SS}( {n,{m + 1}} )} = {\frac{1}{L - n}{\sum\limits_{k = {1 - {NL}}}^{- n}{\underset{\_}{s}*( {k + m + 1} )\mspace{11mu}\underset{\_}{s}\mspace{11mu}( {k + n + m + 1} )}}}} \\{= {\frac{1}{L - n}\;{\sum\limits_{k = {2 - L}}^{{- n} + 1}{\underset{\_}{s}*( {k + m} )\mspace{11mu}\underset{\_}{s}\mspace{11mu}( {k + n + m} )}}}}\end{matrix} & (19)\end{matrix}$

Using equation (18) equation (19) can be expressed as:

$\begin{matrix}{{r_{SS}( {n,{m + 1}} )} = {{r_{SS}( {n,m} )} + {\frac{\begin{matrix}{{\underset{\_}{s}*( {m - n + 1} ){\underset{\_}{s}( {m + 1} )}} -} \\{\underset{\_}{s}*( {m + 1 - L} ){\underset{\_}{s}( {1 - L + m + n} )}}\end{matrix}}{L - n}.}}} & (20)\end{matrix}$

The computational effort for calculating such coefficients of theautocorrelation sequence is low and independent from the correlationinfluence length.

Exponential Smoothing of the Autocorrelation Coefficients

The above given expression for coefficients of the autocorrelationsequence requires a memory capable of holding a number of L complexsamples of the signal s. In order to reduce the memory required forcomputing equation (20), an exponential averaging is employed resultingin the following expression:r _(ss)(n, m+1)=α· s *(m−n+1) s (m+1)+(α−1)·r _(ss)(n,m)  (21)

The parameter a is chosen to be 1/L whereby the influence of the“newest” or “latest” product s*(m−n+1)s(m+1) on the result obtained fromequation (21) will be the same as for equation (20).

As a result, the memory required for a signal length is given by n_(max)which is the largest coefficient of the autocorrelation function to beevaluated, which is, in general, much smaller than the correlationinfluence length.

Calculation of Discrete Values of the Correlation Function's Derivatives

The autocorrelation sequence r_(ss)(n) is a time discrete representationof the autocorrelation function φ_(cc)(t):r _(cc)(n)=φ_(cc) nT  (22)wherein T denotes a sampling period. This relation can be employed tocalculate the values of the first and second derivatives of theautocorrelation function. Since the autocorrelation function's firstderivative {dot over (φ)}_(cc)(0) is the slope of the autocorrelationfunction φ_(cc)(t) at a time t=0 and the autocorrelation function'ssecond derivative {umlaut over (φ)}_(cc)(0) is the slope of theautocorrelation function's first derivative {dot over (φ)}_(cc)(t) for atime t=0, the autocorrelation function's first derivative {dot over(φ)}_(cc)(0) can be estimated to be:

$\begin{matrix}{{{{\overset{.}{\varphi}}_{cc}(0)} \approx {{\overset{\sim}{\overset{.}{\varphi}}}_{cc}(0)}} = {\frac{{\varphi_{cc}(T)} - {\varphi_{cc}( {- T} )}}{2T} = {\frac{{r_{cc}(1)} - {r_{cc}( {- 1} )}}{2T}.}}} & (23)\end{matrix}$

Further, some characteristics of the autocorrelation sequence r_(cc)(n)are exploited:

The autocorrelation sequence r_(cc)(n) is a complex expression, whereinthe real part thereof is even such that the following expression isvalid:Re{r _(cc)(k)}=Re{r _(cc)(−k)}  (24)

Moreover, the imaginary part of the autocorrelation sequence r_(cc)(n)is odd and can be defined by:Im{r _(cc)(k)}=−Im{r _(cc)(−k)}  (25)

In addition, the autocorrelation sequence r_(cc)(n) for n=0 representsthe average power of the channel coefficient sequence c(k) (see equation(7)) and is real.

As a result, equation (23) can be written as:

$\begin{matrix}{{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}(0)} = \frac{j\;{Im}\{ {r_{cc}(1)} \}}{T}} & (26)\end{matrix}$

In order to calculate {umlaut over (φ)}_(cc)(0), {umlaut over(φ)}_(cc)(T/2) and {dot over (φ)}_(cc)(−T/2) are calculated:

$\begin{matrix}{{{{{\overset{.}{\varphi}}_{cc}( {- \frac{T}{2}} )} \approx {\overset{\sim}{\overset{.}{\varphi}}( {- \frac{T}{2}} )}} = \frac{{r_{cc}(0)} - {r_{cc}( {- 1} )}}{T}}{and}} & ( {27a} ) \\{{{{\overset{.}{\varphi}}_{cc}( \frac{T}{2} )} \approx {{\overset{\sim}{\overset{.}{\varphi}}}_{cc}( \frac{T}{2} )}} = \frac{{r_{cc}(1)} - {r_{cc}(0)}}{T}} & ( {27b} )\end{matrix}$

This leads to the following expression for {umlaut over (φ)}_(cc)(0):

$\begin{matrix}{{{{\overset{¨}{\varphi}}_{cc}(0)} \approx {{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}(0)}} = {\frac{{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}( \frac{T}{2} )} - {{\overset{\sim}{\overset{.}{\varphi}}}_{cc}( {- \frac{T}{2}} )}}{T} = {\frac{{\varphi_{cc}(T)} - {2{\varphi_{cc}(0)}} + {\varphi_{cc}( {- T} )}}{T^{2}} = {2\frac{{{Re}\{ {r_{cc}(1)} \}} - {r_{cc}(0)}}{T^{2}}}}}} & (28)\end{matrix}$

As a result, the Doppler characteristics of the radio signaltransmission channel are obtained on the basis of only two complexvalues of the autocorrelation sequence namely, r_(cc)(0) and r_(cc)(1)This leads to the following estimations for the Doppler shift B⁽¹⁾ andthe Doppler spread B⁽²⁾:

$\begin{matrix}{{{B^{(1)} \approx {\overset{\sim}{B}}^{(1)}} = {\frac{1}{2\pi\; T}\frac{{Im}\{ {r_{cc}(1)} \}}{r_{cc}(0)}}}{and}} & (29) \\{{B^{(2)} \approx {\overset{\sim}{B}}^{(2)}} = {\frac{1}{2\pi\; T}\sqrt{2 - ( \frac{{Im}\{ {r_{cc}(1)} \}}{r_{cc}(0)} )^{2} - \frac{2{Re}\{ {r_{cc}(1)} \}}{r_{cc}(0)}}}} & (30)\end{matrix}$Error Estimating of the Derivatives of the Autocorrelation Function

In order to estimate the accuracy of the estimations for the first andsecond derivatives for the autocorrelation function, it is assumed thatthe sequence of channel coefficients can be modelled as discrete samplesof a sum of an infinite number of sine signals with random phases andgiven amplitudes:

$\begin{matrix}{{\underset{\_}{c}(t)} = {{\int_{- \infty}^{\infty}{{a(f)}\;{\mathbb{e}}^{j{({{2\;\pi\;{ft}} + {\varphi{(f)}}})}}{\mathbb{d}f}}} = {\int_{- \infty}^{\infty}{{b( {f,t} )}{\mathbb{d}f}}}}} & (31)\end{matrix}$

Since the products of Fourier transformations of the functions b(f₁,t)and b(f₂,t) are zero, these functions are uncorrelated:φ_(b) ₁ _(b) ₂ =F{φ _(b) ₁ _(b) ₂ }  (32)whereinφ_(b) ₁ _(b) ₁ =b ₁(f)*b ₂(−f)resulting in

$\begin{matrix}\begin{matrix}{\phi_{b_{1}b_{2}} = {{F{\{ {b_{1}(t)} \} \cdot F}\{ {b_{2}( {- t} )} \}} \equiv 0}} & ⩔ & {{b_{1}(t)} = {b( {f_{1},t} )}} \\\; & \; & {{b_{2}(t)} = {b( {f_{2},t} )}} \\\; & \; & {f_{1} \neq f_{2}}\end{matrix} & (33)\end{matrix}$

Thus, the cross power spectrum density is equal to zero and, as aresult, the cross-correlation function is also zero.

Therefore, the autocorrelation function of c(t) can be written as aninfinite sum of the autocorrelation functions of b(f,t):

$\begin{matrix}{{\varphi_{cc}(\tau)} = {\int_{- \infty}^{\infty}{{\varphi_{bb}( {\tau,f} )}\;{\mathbb{d}f}}}} & (34)\end{matrix}$resulting in

$\begin{matrix}\begin{matrix}{{\varphi_{bb}( {\tau,f} )} = {\lim\limits_{T->\infty}{\frac{1}{2T}\;{\int_{- T}^{T}{{a(f)}\;{\mathbb{e}}^{- {j{({{2\;\pi\;{ft}} + {\varphi{(f)}}})}}}{a(f)}\;{\mathbb{e}}^{j{({{2\;\pi\;{f{({t + \tau})}}} + {\varphi{(f)}}})}}{\mathbb{d}t}}}}}} \\{= {{a^{2}(f)}\;( {{\cos\; 2\;\pi\; f\;\tau} + {j\mspace{11mu}\sin\; 2\;\pi\; f\;\tau}} )}}\end{matrix} & (35)\end{matrix}$

The real part of the autocorrelation function φ_(cc)(τ) comprises aninfinite sum of cosine functions, while the imaginary part thereofcomprises an infinite sum of sine functions. In view of the derivationrules for sine and cosine functions, the autocorrelation function ofb(f,t) and its first and second derivatives evaluated for τ=0 can bewritten as:φ_(bb)(0,f)=a ^(2(f)){dot over (φ)}_(bb)(0,f)=j2fa ²(f){umlaut over (φ)}_(bb)(0,f)=−4π² f ² a ²(f)  (36)

On the basis of the estimation defined in equation (23) for anestimation of the slope of φ_(cc)(τ) for τ=0 by a linear interpolationbetween φ_(cc)(T) and φ_(cc)(−T), the autocorrelation function for aspectral element b(f,t) can be described by:

$\begin{matrix}{{{\overset{.}{\varphi}}_{bb}( {0,f} )} = {{{j\; 2\;\pi\;{{fa}^{2}(f)}} \approx \frac{j\mspace{11mu}{Im}\{ {\varphi_{bb}( {T,f} )} \}}{T}} = {{j \cdot {a^{2}(f)}}\;\frac{\sin( {2\;\pi\;{fT}} )}{T}}}} & (37)\end{matrix}$

Thus, the term 2πfT is estimated to represent sin(2πfT) for which it isrequired that the sampling period T is larger than the inverse of thefrequency f. The same applies for an estimation of {umlaut over(φ)}_(bb)(0,f). The largest error for a spectral element b(f,t) isobtained for a highest frequency f=f_(max).

Estimation Error Due to Additive White Noise

In the case the received radio signal c(k) is disturbed by additivewhite noise n(k), the resulting received signal or input signal x(k) isgiven by:x (k)= c (k)+ n (k).  (38)

Since the additive white noise n(k) is independent from the radio signalc(k), the autocorrelation sequence r_(xx)(k) of the input signal x(k) isgiven by:r _(xx)(k)=r _(cc)(k)+r _(nn)(k)  (39)and the power density spectrum Φ_(xx)(f) is given by:φ_(xx)(f)=φ_(cc)(f)+φ_(nn)(f)  (40)

In the case the additive noise is additive white noise and the noisepower is N, the autocorrelation function r_(nn)(k) of the additive whitenoise can be expressed as:r _(nn)(k)=Nδ(k).  (41)

As a result, the input signal x(k) and its autocorrelation functionφ_(xx)(f) can be written as:r _(xx)(k)=r _(cc)(k)+Nδ(k)  (42)andφ_(xx)(f)=φ_(cc)(f)+N  (43)

Referring to equations (29) and (30), for a radio signal disturbed byadditive white noise, r_(cc)(0) has to be replaced by r_(xx)(0)resulting in the following definitions for estimations of the Dopplershift {tilde over (B)}⁽¹⁾ and the Doppler spread {tilde over (B)}⁽²⁾:

$\begin{matrix}{{\overset{\sim}{B}}^{(1)} = {{\frac{1}{2\;\pi\; T}\frac{{Im}\{ {r_{xx}(1)} \}}{r_{xx}(0)}} = {\frac{1}{2\;\pi\; T}\frac{{Im}\{ {r_{cc}(1)} \}}{{r_{cc}(0)} + N}}}} & (44) \\{and} & \; \\{{\overset{\sim}{B}}^{(2)} = {\frac{1}{2\;\pi\; T}\sqrt{2 - ( \frac{{Im}\{ {r_{xx}(1)} \}}{r_{xx}(0)} )^{2} - \frac{2\mspace{11mu}{Re}\{ {r_{xx}(1)} \}}{r_{xx}(0)}}}} & (45) \\{\mspace{40mu}{= {\frac{1}{2\;\pi\; T}\sqrt{2 - ( \frac{{Im}\{ {r_{cc}(1)} \}}{{r_{cc}(0)} + N} )^{2} - \frac{2\mspace{11mu}{Re}\{ {r_{cc}(1)} \}}{{r_{cc}(0)} + N}}}}} & \; \\{\mspace{40mu}{= {\frac{1}{2\;\pi\; T}\sqrt{2 - \frac{{{Im}^{2}\{ {r_{cc}(1)} \}} - {2\mspace{11mu}{Re}\{ {r_{cc}(1)} \}( {{r_{cc}(0)} + N} )}}{( {{r_{cc}(0)} + N} )^{2}}}}}} & \;\end{matrix}$

As can be derived from equations (44) and (45), it will be appreciatedthat additional white noise has a significant effect on the estimationresults, in particular for the Doppler spread {tilde over (B)}⁽²⁾.

One approach is based on the assumption that a good estimation is givenfor the noise power N for example provided by a signal processing unit.Then, it is possible to simply deduct the noise power N from r_(xx)(0)to compensate the influence of additive white noise. Nevertheless, thisapproach does not take into account the high noise sensitivity of anestimation for the autocorrelation function's second derivative {umlautover (φ)}_(xx)(0) as will be discussed below.

Another approach which is based on estimations without using r_(xx)(0)will be detailed in the following section.

Modifications for Decreasing Noise Sensitivity

As can be derived from equations (41) and (42), additional white noiseonly changes one single value in the autocorrelation sequence φ_(cc) ofthe received-signal c. In particular, the value influenced by additionalwhite noise is the value r_(xx)(0). For further values of the inputsignal's autocorrelation function r_(xx)(k) with k≠0 the autocorrelationsequence remains unchanged and is thus independent from additional whitenoise.

Further, it has to be considered that the above relation can beconsidered valid for an infinite correlation influence length L of theautocorrelation function. Here, the autocorrelation function r_(nn)(k)of the additive white noise can be estimated by:

$\begin{matrix}{{r_{nn}(k)} = \{ \begin{matrix}{N,} & {k = 0} \\{\frac{N}{\sqrt{L}},} & {k \neq 0}\end{matrix} } & (46)\end{matrix}$

For calculating the Doppler parameters without using r_(xx)(0), it isreferred to equations (44) and (45) wherein r_(xx)(0) appears in thedenominator. On the basis of the above results (in particular seesection “error in estimating the derivatives”) the autocorrelationfunction φ_(cc)(0) for a received radio signal can be estimated by:φ_(cc)(0)={tilde over (φ)}_(cc) ⁽¹⁾(0)=Re{r _(xx)(1)}  (47)

For an estimation of the first derivative of the autocorrelationfunction {dot over (φ)}_(cc)(0) (see equation (26)), φ_(cc)(0) is notemployed which results in:

$\begin{matrix}{{{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}(0)} \approx {{\overset{\sim}{\overset{.}{\varphi}}}_{cc}^{(1)}(0)}} = \frac{j\mspace{11mu}{Im}\{ {r_{xx}(1)} \}}{T}} & (48)\end{matrix}$

In equation (26), the second derivative of the autocorrelation function{dot over (φ)}_(cc)(0) was estimated on the basis of the differencebetween {dot over (φ)}_(cc)(−T/2) and {umlaut over (φ)}_(cc)(T/2). Here,this estimation is based on the difference between {dot over(φ)}_(cc)(1, 5T) and {dot over (φ)}_(cc)(−1, 5T) leading to:

$\begin{matrix}\begin{matrix}{{{\overset{¨}{\varphi}}_{cc}(0)} \approx \frac{{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}( {1.5T} )} - {V\;{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}( {{- 1.5}T} )}}}{3T}} \\{= \frac{{r_{cc}(2)} - {r_{cc}(1)} - ( {{r_{cc}( {- 1} )} - {r_{cc}( {- 2} )}} )}{3T^{2}}} \\{= {2\frac{{Re}\{ {{r_{cc}(2)} - {r_{cc}(1)}} \}}{3T^{2}}}}\end{matrix} & (49)\end{matrix}$

From equation (49), an estimation for the disturbed input signal x canbe defined as:

$\begin{matrix}{{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{(1)}(0)} = {2{\frac{{Re}\{ {{r_{xx}(2)} - {r_{xx}(1)}} \}}{3T^{2}}.}}} & (50)\end{matrix}$

The estimation of the slope of the first derivative of theautocorrelation function φ_(cc) at time 0 is based on linearapproximation between two points being separated by a duration of 3T.Therefore, it has to be considered that this approximation will providebetter results the more the condition 3Tf<<1 is fulfilled (see section“error in estimating the derivatives”).

On the basis of the above observations, estimations for the Dopplershift {tilde over (B)}⁽¹⁾ and the Doppler spread {tilde over (B)}⁽²⁾ aregiven by:

$\begin{matrix}{{\overset{\sim}{B}}_{2}^{(1)} = {\frac{1}{2\pi\; T}\mspace{11mu}\frac{{Im}\{ {r_{xx}(1)} \}}{{Re}\{ {r_{xx}(1)} \}}\mspace{14mu}{and}}} & (51) \\{{\overset{\sim}{B}}_{2}^{(2)} = {\frac{1}{2\pi\; T}\sqrt{\frac{2}{3} - ( \frac{{Im}\{ {r_{xx}(1)} \}}{{Re}\{ {r_{xx}(1)} \}} )^{2} - \frac{2\mspace{11mu}{Re}\{ {r_{xx}(2)} \}}{3\mspace{11mu}{Re}\{ {r_{xx}(1)} \}}}}} & (52)\end{matrix}$

Calculating this more general expressions based on r_(xx)(l) andr_(xx)(k) for k≧2 leads to:

$\begin{matrix}{{{\overset{\sim}{\varphi}}_{cc}^{(\kappa)}(0)} = {{Re}\{ {r_{xx}(1)} \}}} & (53) \\{{{\overset{\overset{\sim}{.}}{\varphi}}_{cc}^{(\kappa)}(0)} = \frac{j\;{Im}\{ {r_{xx}(1)} \}}{T}} & (54) \\{{{\overset{\overset{\sim}{.}}{\varphi}}_{cc}^{(\kappa)}( \frac{\kappa + 1}{2} )} = \frac{{r_{xx}(\kappa)} - {r_{xx}(1)}}{( {\kappa - 1} )T}} & (55) \\{{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{(\kappa)}(0)} = {\frac{{{\overset{\sim}{\overset{.}{\varphi}}}_{cc}^{\kappa}( \frac{\kappa + 1}{2} )} - {{\overset{\sim}{\overset{.}{\varphi}}}_{cc}^{\kappa}( {- \frac{\kappa + 1}{2}} )}}{( {\kappa + 1} )T} = {2\frac{{Re}\{ {{r_{xx}(\kappa)} - {r_{xx}(1)}} \}}{( {\kappa^{2} - 1} )T^{2}}}}} & (56)\end{matrix}$

As a result, the Doppler shift B⁽¹⁾ and the Doppler spread B⁽²⁾ can beestimated as follows:

$\begin{matrix}{{\overset{\sim}{B}}_{\kappa}^{(1)} = {\frac{1}{2\pi\; T}\mspace{11mu}\frac{{Im}\{ {r_{xx}(1)} \}}{{Re}\{ {r_{xx}(1)} \}}\mspace{14mu}{and}}} & (57) \\{{\overset{\sim}{B}}_{\kappa}^{(2)} = {\frac{1}{2\pi\; T}\sqrt{\frac{2}{\kappa^{2} - 1} - ( \frac{{Im}\{ {r_{xx}(1)} \}}{{Re}\{ {r_{xx}(1)} \}} )^{2} - \frac{2\mspace{11mu}{Re}\{ {r_{xx}(\kappa)} \}}{( {\kappa^{2} - 1} )\mspace{11mu}{Re}\{ {r_{xx}(1)} \}}}}} & (58)\end{matrix}$Choosing the Optimal Autocorrelation Coefficient

The results of the preceding sections allow for a further control ofcontrol on the noise sensitivity of the estimations for the Dopplerparameters.

Under noisy conditions, assuming k<<L, equation (56) results in:

$\begin{matrix}{{{\overset{\sim}{\overset{¨}{\varphi}}}_{\underset{\sim}{cc}}^{\kappa}(0)} = {2\frac{{{Re}\{ {{r_{cc}(\kappa)} - {r_{cc}(1)}} \}} + \frac{N}{\sqrt{L}}}{( {\kappa^{2} - 1} )T^{2}}}} & (59)\end{matrix}$

The relative error ε_(noise) due to additive white noise can becalculated by:

$\begin{matrix}{ɛ_{noise} = {\frac{{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)} - {{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)}}{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)} = \frac{N}{\sqrt{{L \cdot {Re}}\{ {{r_{cc}(\kappa)} - {r_{cc}(1)}} \}}}}} & (60)\end{matrix}$

As detailed in section “error in estimating the derivatives”, r_(cc)(k)has a cosine like shape for a band limited signal whereink·T<1/(f_(max)). The absolute value of the denominator will increase fora raising k leading to an increase of the estimation accuracy asillustrated in FIG. 2 calculated for a signal to noise ratio of 1 andL=15.000.

In order to calculate the maximum error due to the linear approximationof {umlaut over (φ)}_(cc)(0), the worst case power distribution isassumed wherein power is concentrated at f=f_(max). This results in thepower density spectrum:

$\begin{matrix}{{\phi_{ss}(f)} = {\frac{S}{2}( {{\delta( {f + f_{\max}} )} + {\delta( {f - f_{\max}} )}} )}} & (61)\end{matrix}$where S denotes the signal power.

Equation (57) can be transformed to the autocorrelation functionφ_(xx)(t) and the second derivative {umlaut over (φ)}_(xx)(t) for t=0:φ_(xx)(t)=S·cos(2πf _(max) t)  (62)and{umlaut over (φ)}_(xx)(0)=−4π² f ² _(max) S  (63)

The estimation for {umlaut over (φ)}_(xx)(0) results to:

$\begin{matrix}{{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)} = {{2\frac{{Re}\{ {{\varphi_{cc}(\kappa)} - {\varphi_{cc}(1)}} \}}{( {\kappa^{2} - 1} )T^{2}}}\mspace{65mu} = {2S\frac{{\cos( {2\pi\; f_{\max}\kappa\; T} )} - {\cos( {2\pi\; f_{\max}T} )}}{( {\kappa^{2} - 1} )T^{2}}}}} & (64)\end{matrix}$

This leads to the following definition of an estimated relative errorε_(approx):

$\begin{matrix}{ɛ_{approx} = {\frac{{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)} - {{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)}}{{\overset{\sim}{\overset{¨}{\varphi}}}_{cc}^{\kappa}(0)}\mspace{70mu} = {{\frac{1}{2}\mspace{11mu}\frac{{\cos( {2\pi\; f_{\max}T} )} - {\cos( {2\pi\; f_{\max}\kappa\; T} )}}{( {\kappa - 1} )T^{2}\pi^{2}f_{\max}^{2}}} - 1}}} & (65)\end{matrix}$

A graph illustrating the thus obtained estimated relative errorε_(approx) is shown in FIG. 3. Comparing FIGS. 2 and 3, it can beappreciated that a certain κ has to be chosen to keep both the relativeerror ε_(noise) and the approximated relative error ε_(approx) below apredefined error ε_(max).

A summed error ε_(sum) obtained by an addition of the amount of themaximal relative error ε_(noise) and the amount of the maximalapproximated relative error ε_(approx) is illustrated in FIG. 4. FromFIG. 4 it can be derived that a specific κ has to be chosen fordifferent Doppler spreads which will be discussed below in greaterdetail.

Influence of the Slot Format

As explained in the beginning, the approach to estimate the Dopplerparameters on the basis of pilot symbols or groups does not result ingood estimations for a Doppler spectrum for which high frequencies areexpected. In order to provide for a sufficient estimation of Dopplerparameters even for high frequency offsets, the solution describedherein is employed on symbol level, particularly by using demodulatedpilot symbols of a pilot group contained in a slot. As a result, thesampling period T utilized here essentially corresponds to the symbolrate in a slot. For example, in an UMTS system, the symbol rate for anuplink data communication is 15 kHz leading to a sampling period T of6,6·10⁻⁵ seconds. Thus, in an UMTS system frequencies of up to 7.500 Hzcan be estimated.

As shown above, a calculation of an autocorrelation function whichrepresents the only calculation on sampling rate T does not requirecomplex computations and further involves computations only of a smallnumber of coefficients.

In the following, the symbol level solution is described with respect toa mobile communications environment according to the standards of GSM orUTMS. Therefore, some issues discussed below will not araise in othercommunications environments wherein the Doppler shift and Doppler spreadestimation presented herein is used and which are not operated accordingto the GSM or UTMS standard.

As shown in FIG. 5, the so called dedicated physical control channelDPCCH is used to transmit radio frames including a predefined number ofslots. Each slot is comprised by a predefined number of pilot bitshaving a predefined pattern, feedback information (FBI) bits, bits fortransmit power control (TPC) commands and optionally bits for atransport format combination indicator (TFCI). The pilot bits or pilotsymbols which are used for an estimation of Doppler shift and Dopplerspread of subsequent slots do not represent a continuous sequence butare separated by other control bits such as TPC, FBI and FTCI bits.

Thus, each slot includes a number of pilot bits n_(p) followed by anumber of n_(s)−n_(p) other control bits resulting in number of n_(s)bits per slot. To extract a sequence of pilot bits of slots of a commonframe a pilot sequence is defined as a sequence of n_(p) pilot bits forthe first slot followed by n_(s)−n_(p) zeros followed by a sequence ofn_(p) pilot bits of a second slot followed by n_(s)−n_(p) zeros followedby a sequence of n_(p) pilot bits of a third slot or followed byn_(s)−n_(p) zeros, . . . and so forth for all slots of a frame. Assuminga received signal y(n) demodulated with such a pilot sequence, ademodulated signal y(k) shown in FIG. 6 will result.

For illustrating a calculation of the autocorrelation coefficientsr_(yy)(0) and r_(yy)(1) it is referred to FIG. 7. This figure shows acalculation of a specific autocorrelation coefficient r_(xx)(m) of theautocorrelation sequence of a time discrete signal x(n) comparable tothe so called “Papierstreifenmethode”. The first input sequence isconjugated and written in the first line. The input sequence is shiftedby m and written in the second line. Then, the values of same columnsare multiplied (for example: x*(n)·x(n+m) and each of the results isrespectively written in the same column. As indicated by the expressionbelow the thick line in FIG. 7, the resulting products are added and theresulting sum is divided by the number of columns. This results in them^(th) coefficient of the autocorrelations.

In a comparable manner, FIG. 8 illustrates a calculation of theautocorrelation coefficient r_(yy)(0) of the demodulated signal y(k). Itcan be seen that for n_(s)−n_(p) columns per slot the multiplications ofrespective values of the first and second lines result in a value of 0and that n_(p) columns per slot are left. As a result, the averagingprocess for the interrupted sequence shown in FIG. 6 is lowered by afactor n_(p)/n_(s) compared to the averaging process illustrated in FIG.7. The autocorrelation coefficient r_(xx)(0) can be written as:

$\begin{matrix}{{r_{xx}(0)} = {\frac{n_{s}}{n_{p}}{r_{yy}(0)}}} & (66)\end{matrix}$

This is due to the fact that the coefficient “0” of an autocorrelationfunction characterizes the respective signal power, wherein the power ofthe demodulated signal y(k) is reduced by the factor n_(p)/n_(s).

For a calculation of coefficient r_(yy)(1), as illustrated in FIG. 9, anumber of n_(s)−n_(p)+1 columns having a value of zero is obtained perslot. Thus, the averaging process results in a reduction by a factor

$\frac{n_{s}}{n_{p} - 1}:$

$\begin{matrix}{{r_{xx}(1)} = {\frac{n_{s}}{n_{p} - 1}{r_{yy}(1)}}} & (67)\end{matrix}$

Performing a comparable calculation for the autocorrelation coefficientr_(cc)(2), the following expression can be obtained:

$\begin{matrix}{{r_{xx}(2)} = {\frac{n_{s}}{n_{p} - 2}{r_{yy}(2)}}} & (68)\end{matrix}$

Here, the reducing factor is

$\frac{n_{s}}{n_{p} - 2}.$

This leads to a re-construction of the autocorrelation sequencer_(xx)(n) of the non-interrupted signal x(k) whereby the methoddescribed therein can be based on calculations on symbol or bit level.In general, the autocorrelation sequence r_(xx) can be defined by

$\begin{matrix}{{{r_{xx}( {{k \cdot n_{s}} + m} )} = {\frac{n_{s}}{n_{s} - m}{r_{yy}( {{k \cdot n_{s}} + m} )}}}\{ \begin{matrix}{\kappa\mspace{14mu}{an}\mspace{14mu}{integer}} \\{0 \leq m < {n_{p^{\prime}}m} \leq {n_{s} - n_{p}}}\end{matrix} } & (69)\end{matrix}$Doppler Spread and Speed of a Mobile Entity

In order to modify the adaptation speed of transmission and/or receivingmeans (hardware and software) of a moving mobile entity it can berequired to determine the speed of motion for the mobile entity. In acommunications environment wherein radio signals are only transmittedvia single line of sight channels, the Doppler shift or Dopplerfrequency f_(d) resulting from a moving entity allows for a calculationof the speed of the entity according to equation of (1). In contrastthereto, as can be derived from the definition of the Doppler spread inequation (11) this is, in general, not possible for a multipathpropagation environment. In particular, the Doppler spread as defined inequation (11) does not exhibit a maximum spread frequency f_(max).

Rather, a definition of a maximum spread frequency f_(max) requires acertain shape of a Doppler power density spectrum:

Assuming a Jake's spectrum, as set forth above, solving equation (11)using equation (6) results in :

$\begin{matrix}{B_{J}^{(2)} = \frac{f_{\max.}}{\sqrt{2}}} & (70)\end{matrix}$

For a uniform power distribution between two frequencies−f_(max)<f<f_(max) expressed as

$\begin{matrix}{{\phi(f)} = {\frac{1}{2f_{\max}} \cdot {{rect}( \frac{f}{f_{\max}} )}}} & ( {72a} )\end{matrix}$the Doppler spread B⁽²⁾ is given by:

$\begin{matrix}{B_{U}^{(2)} = \frac{f_{\max}}{\sqrt{3}}} & ( {72b} )\end{matrix}$

In the case of a spectrum only exhibiting components at f_(max) and−f_(max), as described by equation (61), the result is:B ⁽²⁾ =f _(max)  (72)

As can be derived from the above examples, the more power isconcentrated at edges of the power spectrum the more the factor r givenby

$\begin{matrix}{r = \frac{B^{(2)}}{f_{\max}}} & (73)\end{matrix}$tends to be 1.

Further, if all power is concentrated in the center of the powerspectrum, the Doppler spread will become 0. This applies also for asuperposition of a Doppler power density resulting from radio signaltransmitted via direct and indirect paths. For an estimation of themaximum frequency f_(max), a factor r has to be chosen in the range of0<r<1. Choosing the factor r to be 1/√{square root over (2)}, the speedof a mobile entity can be expressed as:

$\begin{matrix}{v = {\frac{f_{\max}c_{0}}{f_{c}} \approx {\frac{B^{(2)}}{r}\mspace{11mu}\frac{c_{o}}{f_{c}}}}} & (74)\end{matrix}$wherein c₀ denotes the speed of light and f_(c) denotes the carrierfrequency.Estimation of the Doppler Parameters

On the basis of the result described above, it is possible to define analgorithm for estimations of Doppler parameters. For a calculation ofautocorrelation coefficients, at least three different lags κ arerequired to keep an estimation error for the second derivative of theautocorrelation function below 10%, as illustrated in FIG. 4. Furtherreferring to FIG. 4, three lags κ are chosen to be 2, 10 and 100. Inaddition, the lags 0 and 1 are employed resulting in a calculation offive coefficients of the autocorrelation sequence.

A calculation of autocorrelation coefficients can be based on equation(20) or, to be more efficient, on equation (21). Referring to therespective part of the description, the choice of a correlationinfluence length L depends on the worst signal to noise ratio which isexpected to occur or which shall be considered. Assuming a signal tonoise ratio of 0 dB, the influence length L of 15.000 is chosen wherebya duration of 1 second is averaged. This has to be performed with thesymbol or bit rate which is 15 kHz in case of an UMTS system.

A compensation of the slot format will be performed for the calculatedfive autocorrelation coefficients according to equation (69).

The compensation for the slot format and all further calculations haveto be performed as often an update of the Doppler parameters isrequired.

For a calculation of the Doppler parameters, a suitable lag κ has to befound in order to keep the estimation errors low or under a predefinedthreshold. One solution is to search for a lag κ for which the estimatedsignal to noise ratio SNR_(lag) is above a predefined threshold signalto noise ratio SNR_(thr). The smallest lag κ will result in the worstsignal to noise ratio for signals with a small Doppler spread while thebest estimation error is obtained for signal having higher Dopplerspreads. Therefore, the definition for a suitable lag κ can be startedwith the smallest lag κ.

On the basis of equation (42), the noise power N is calculated:N=Re{φ _(xx)(0)}−Re{φ _(xx)(1)}  (75)

The estimation of a noise influence in a specific lag κ is given by:

$\begin{matrix}{N_{lag} = \frac{N}{\sqrt{L}}} & (76)\end{matrix}$which is the same for all lags κ.

According to equation (61) the estimation of the second derivative ofthe autocorrelation function is based on the undisturbed signal.Comparing equation (61) and (59) results in:

$\begin{matrix}{{{Re}\{ {{r_{xx}(\kappa)} - {r_{xx}(1)}} \}} \approx {{{Re}\{ \;{{r_{cc}(\kappa)} - {r_{cc}(1)}} \}} + \frac{N}{\sqrt{L}}}} & (77)\end{matrix}$

Then, the estimated signal to noise ratio SNR_(lag) can be calculated onthe basis of the following expression:

$\begin{matrix}{{SNR}_{lag} = {\frac{{Re}\{ {{r_{cc}(\kappa)} - {r_{cc}(1)}} \}}{N_{lag}} \approx {\frac{{Re}\{ {{r_{xx}(\kappa)} - {r_{xx}(1)}} \}}{N_{lag}} - 1}}} & (78)\end{matrix}$

This leads to the following algorithm.

-   κ=2-   threshold=(SNR_(min)+1)·N/√{square root over (L )}-   while (φ_(xx)(1)−φ_(xx)(κ)<threshold && κ!=not found)-   κ=next κ

The above algorithm iterates through the lags κ in ascending orderassuming that the signal to noise ratio increases with an increasinglag. In the case no lag is found either the noise is too high toestimate a Doppler spread or the Doppler spread is too small. Then it ispossible to set the Doppler spread to zero or to calculate a Dopplerspread on the basis of equation (58).

Then, the speed of a moving entity is estimated by multiplying theDoppler spread with a factor that depends on the geometry. As anexample, the factor is set to √{square root over (2)} as described insection “Doppler spread and speed of a mobile entity”. The resultingmaximum spread frequency f_(max) is multiplied with c₀/f₀.

For a calculation of the Doppler shift equation (26) can be used.

Simulation Environment

The algorithm performance was investigated by simulations. The abovealgorithms and the required signal sources have been implemented incomputer program and a structure shown in FIG. 10 was used. As inputsignal for the algorithms, a combined signal consisting of

-   -   a direct signal,    -   an indirect signal, and    -   a noise signal        was used. The direct signal was implemented as a complex sine        signal:        c _(d)(k)=a _(d) e ^(j2π) ^(f) d ^(κT) =a _(d) cos(2πf _(d)        kT)+j·a _(d) sin(2πf _(d) kT)  (79)        wherein a_(d) denotes the amplitude of the direct signal and        f_(d) the frequency offset of the direct path. a_(d) was        expressed as the root of the direct signal's power P_(d):        a _(d) =√{square root over (P _(d) )}  (80)

The indirect signal was modelled to have a power density spectrumaccording to equation (6), a Jake's spectrum. The indirect signal wasmodeled as sum of two independent noise processes μ₁ and μ₂:C _(J)(k)=μ₁(k)+jμ ₂(k)  (81)

A noise process can be modeled as sums of sine functions of differentfrequencies and uniformly distributed random phases φ_(i,n):

$\begin{matrix}{{\mu_{i}(k)} = {\sum\limits_{n = 1}^{Ni}\;{V_{i,n}{\sin( {{2\pi\; f_{i,n}{kT}} + \varphi_{i,n}} )}}}} & (82)\end{matrix}$

Parameters in equation (82) are calculated as:

$\begin{matrix}{{v_{i,n} = {\sigma_{i}\sqrt{\frac{2}{N_{i}}}}}{and}} & (83) \\{f_{i,n} = {f_{\max}{\sin( {\frac{\pi}{2N_{i}}( {n - \frac{1}{2}} )} )}}} & (84)\end{matrix}$wherein σ_(i) denotes the variance of the noise process μ_(i). The powerP_(i) of the indirect signal can be expressed by P_(i)=σ₁ ²+σ₂ ².Choosing σ₂=σ₁ results in:

$\begin{matrix}{\sigma_{i} = \sqrt{\frac{P_{i}}{2}}} & (85)\end{matrix}$

Achieving independent processes μ₁ and μ₂ can be accomplished by settingN₂=N₁+1. The model results in a good approximation of theautocorrelation function of the process up to the N₁ ^(th) zero crossingof the autocorrelation function. To improve the simulation speed, therequired sine signals were generated as set forth below.

Since the simulations require the calculation of many different sinesignals, these signals are not generated using the generic “sin”function. Instead-recursive calculations have been performed using ashift register.

For a z-transformation, the following definitions have been used:

$\begin{matrix}{ {\sin( {2\pi\;{fkT}} )}arrow\frac{z\;{\sin( {2\pi\;{fT}} )}}{z^{2} - {2z\;{\cos( {2\pi\;{fT}} )}} + 1} {and}} & (86) \\{{\cos( {2\pi\;{fkT}} )}->\frac{z( {z - {\cos( {2\pi\;{fT}} )}} )}{z^{2} - {2z\;{\cos( {2\pi\;{fT}} )}} + 1}} & (87)\end{matrix}$

An interpretation of equations (86) and (87) as impulse response to asignal given by:

$\begin{matrix}{{x(k)} = \{ \begin{matrix}{1,{k = 0}} \\{0,{k \neq 0}}\end{matrix} } & (88)\end{matrix}$leads to the following relations for the sine signal:y(k)=sin(2πfkT)=x(k−1)sin(2πfT)+2y(k−1)cos(2πfT)−y(k−2)  (89)andy(k)=cos(2πfkT)=x(k)+cos(2πfkt)x(k−1)+2y(k−1)cos(2πfT)−y(k−2)  (90)

An expression for a phase shifted sine signal can be derived by usingthe following equation:sin(a+b)=sin(a)cos(b)+cos(a)sin(b)  (91)

Setting a=2πfkT and b=φ and transforming the result in the z-domainusing equations (86) and (87) leads to:

$\begin{matrix}{{\sin( {{2\pi\;{fkT}} + \varphi} )}->\frac{{z^{2}\sin\;\varphi} + {z( {{{\sin( {2\pi\;{fT}} )}\cos\;\varphi} - {{\cos( {2\pi\;{fT}} )}\sin\;\varphi}} )}}{z^{2} - {2z\;{\cos( {2\pi\;{fT}} )}} + 1}} & (92)\end{matrix}$and finally to

$\begin{matrix}\begin{matrix}{{y(k)} = {\sin( {{2\pi\;{fkT}} + \varphi} )}} \\{= {{{{x(k)} \cdot \sin}\;\varphi} + {{x( {k - 1} )} \cdot ( {{{\sin( {2\pi\;{fT}} )}\cos\;\varphi} -} }}} \\{ {{\cos( {2\pi\;{fT}} )}\sin\;\varphi} ) + {2{{y( {k - 1} )} \cdot {\cos( {2\pi\;{fT}} )}}} - {y( {k - 2} )}}\end{matrix} & (93)\end{matrix}$

Note that all multiplication factors are constant values and only haveto be calculated once when the simulation starts.

A frequency shift of the indirect signal is obtained by multiplexing itwith a complex sine signal. The noise signal is generated as a sum oftwo independent Gaussian noise processes with the same variance for thereal and imaginary component.

Simulation Results for an Indirect Non Shifted Signal

In this simulation, no direct path signal is generated and the Dopplershift is set to zero. The parameter is the Doppler spread which appearsat different mobile speeds in this particular environment. Results areshown in FIG. 11.

Simulation Results for a Direct Shifted Signal

This simulation is performed without any indirect signal so the receivedsignal only consists of a direct path components and noise. Theparameter is the frequency shift. Results are shown in FIG. 12.

Simulation Results for a Superposed Direct and Indirect Signal

This simulation assumes a superposition of a direct path and a reflectedcomponent of the same power. The reflected component has no Dopplershift and a Doppler spread of 200 Hz. The indirect path's Doppler shiftis used as parameter in this simulation and is the range from 0 to 200Hz. The existence of a direct component decreases the estimation of theDoppler shift to a certain degree, in particular when the directcomponent appears in the center of the spectrum of the reflectedcomponent. Results are shown in FIG. 13.

1. A method for Doppler spread estimation for a radio signaltransmission channel in a mobile communications environment on the basisof a radio signal (c) transmitted via the transmission channel, themethod comprising the steps of: determining an autocorrelation function(φ_(cc)(t)) for the radio signal (c); defining a Doppler spread (B⁽²⁾)for the radio signal (c) as a function in time of the autocorrelationfunction (φ_(cc)(t)) and its first and second derivatives for a point oftime being zero; determining the autocorrelation function (φ_(cc)(0))for the point of time being zero; estimating the first derivative of theautocorrelation function (φ_(cc)(0)) for the point of time being zero,by averaging a first portion of the autocorrelation function (φ_(cc)(t))including the point of time being zero; estimating the second derivativeof the autocorrelation function (φ_(cc)(0)) for the point of time beingzero by averaging a second portion of the autocorrelation function(φ_(cc)(t)) including the point of time being zero; and estimating theDoppler spread (B⁽²⁾) for the transmission channel by evaluating theDoppler spread function using the autocorrelation function (φ_(cc)(0))as determined for the point of time being zero and the first and secondderivatives (φ_(cc)(0), (φ_(cc)(0)) of the autocorrelation function asestimated for the point of time being zero.
 2. The method according toclaim 1, wherein determining of the autocorrelation function (φ_(cc)(t))comprises the steps of: modelling the radio signal (c) as a timediscrete signal; and determining an autocorrelation sequence (r_(cc)(n))on the basis of the time discrete signal characterizing the radio signal(c).
 3. The method according to claim 2, wherein the step of determiningthe autocorrelation function (φ_(cc)(t)) comprises the steps of:defining a correlation influence length (L) for the autocorrelationfunction (φ_(cc)(t)); and determining a recursive function forrecursively determining coefficients of the autocorrelation sequence(r_(cc)(n)).
 4. The method according to claim 3, wherein the step ofdetermining the recursive function comprises the step of: determiningthe recursive function by exponentially averaging of the autocorrelationsequence (r_(cc)(n)).
 5. The method according to claim 2, wherein, forthe step of estimating the first and second derivatives (φ_(cc)(0),(φ_(cc)(0)) of the autocorrelation function, discrete autocorrelationcoefficients of the autocorrelation sequence (r_(cc)(n) are used.
 6. Themethod according to claim 5, wherein, for the step of estimating thefirst and second derivatives (φ_(cc)(0), (φ_(cc)(0)) of theautocorrelation function, a first autocorrelation coefficient (r_(cc)⁽¹⁾) and a second autocorrelation coefficient (r_(cc)(K)) of theautocorrelation sequence (r_(cc)(u) are used.
 7. The method according toclaim 6, wherein the second autocorrelation coefficient (r_(cc)(K)) isdetermined in dependence of the correlation influence length (L).
 8. Themethod according to claim 6, further comprising the steps of: defining athreshold value (SNR_(thr)) for a signal to noise ratio; defining asignal to noise ratio (SNR_(lag)) for the estimated second derivative(φ_(cc)(t)) of the autocorrelation function; determining the signal tonoise ratio (SNR_(lag)) for the estimated second derivative (φ_(cc)(t))of the autocorrelation function on the basis of autocorrelationcoefficients (r_(cc)(1), r_(cc)(K)) for the autocorrelation sequence(r_(cc)(n); and determining an autocorrelation coefficient as the secondautocorrelation coefficient (r_(cc)(K) for which the determined signalto noise ratio (SNR_(lag)) is below the defined signal to noise rationthreshold value (SNR_(thr)).
 9. The method according to claim 1, whereinthe step of estimating the first derivative (φ_(cc)(0)) of theautocorrelation function for the point of time being zero comprises thestep of: determining the slope of the autocorrelation function(φ_(cc)(t)) for the point of time being zero.
 10. The method accordingto claim 9, wherein determining the slope of the autocorrelationfunction (φ_(cc)(t)) comprises the steps of: defining a first point oftime (−1, 5T; −T; −T/2) being negative; defining a second point of time(T/2; T; 1,5T) being positive; determining respective values of theautocorrelation function (φ_(cc)(t)) for the first and second points oftime; and determining a respective one of the slopes of theautocorrelation function (φ_(cc)(t)) on the basis of the determinedrespective values of the autocorrelation function (φ_(cc)(t)) for thefirst and second points of time in relation to a time interval definedby the first and second points of time.
 11. The method according toclaim 10, further comprising the step of: defining the first and secondpoints of time respectively for each slope of the autocorrelationfunction (φ_(cc)(t)) used for the estimating of its first and secondderivatives (φ_(cc)(0), (φ_(cc)(0)) for the point of time being zerosuch that the estimating of the first and second derivatives (φ_(cc)(0),(φ_(cc)(0)) of the autocorrelation function is based on two values ofthe autocorrelation function (φ_(cc)(t)), whereby the estimating of theDoppler spread (B⁽²⁾) for the transmission channel is performed by usingthe two values of the autocorrelation function (φ_(cc)(t)).
 12. Themethod according to claim 1, wherein the step of estimating the secondderivative (φ_(cc)(0)) of the autocorrelation function for the point oftime being zero comprises the step of: determining the slope of thefirst derivative (φ_(cc)(t)) of the autocorrelation function for thepoint of time being zero, wherein values of the first derivative(φ_(cc)(t)) of the autocorrelation function used for the estimating ofthe slope thereof are determined by determining the slope of respectiveportions of the autocorrelation function (φ_(cc)(t)).
 13. The methodaccording to claim 1, wherein the steps of estimating the first andsecond derivatives (φ_(cc)(0), (φ_(cc)(0)) of the autocorrelationfunction each include the step of: estimating on the basis of portionsof the autocorrelation function (φ_(cc)(t)) being unaffected by signalnoise (n) of the radio signal (c).
 14. The method according to claim 13,further comprising the step of: estimating the first and secondderivatives (φ_(cc)(0), (φ_(cc)(0)), of the autocorrelation function forthe point of time being zero on the basis of values for theautocorrelation function (φ_(cc)(t)) not including the value of theautocorrelation function (φ_(cc)(0)) for the point of time being zero.15. The method according to claim 1, further comprising the steps of:receiving the radio signal (c); demodulating the received radio signal(c) with a predefined signal sequence (n_(p), n_(s)); and determiningthe autocorrelation function (φ_(cc)(t)) as autocorrelation for thedemodulated radio signal (y).
 16. The method according to claim 15,wherein the autocorrelation function (φ_(yy)(t)) for the demodulatedradio signal (y) is defined by an autocorrelation sequence r_(yy)(n) forthe demodulated radio signal (y).
 17. The method according to claim 16,wherein the autocorrelation sequence is defined by an autocorrelationsequence (r_(xx)(n)) for demodulated radio signal portions (x)representing the predefined signal sequence (n_(p),n_(s)).
 18. Themethod according to claim 17, wherein autocorrelation coefficients forthe demodulated radio signal portions (x) are determined on the basis ofan averaging process for respective autocorrelation coefficients(r_(yy)(n)) for the demodulated radio signal (y) by means of recursivefunction.
 19. The method according to claim 18, wherein the averagingprocess is performed in dependence on a percentage of the predefinedsignal sequence (u_(p),u_(s)) with respect to the de-modulated radiosignal (y).
 20. The method according to claim 1, wherein thetransmission channel is a DPCCH channel of a mobile telephoneenvironment, wherein the radio signal (c) comprises at least one framehaving subsequent slots each thereof including a number of predefinedpilot symbols, the method further comprising the steps of: demodulatingthe radio signal (c) to obtain demodulated pilot symbols per slots(x(n_(p))); determining the autocorrelation function on the basis ofautocorrelation coefficients (r_(xx)(n)) of anautocorrelation sequencedetermined for the demodulated pilot symbols (x(u_(p))); estimating thefirst and second derivatives (φ_(cc)(t), (φ_(cc)(t)) of theautocorrelation function in dependence, of autocorrelation coefficients(r_(xx)(n) for the demodulated pilot symbols (x(n_(p))); and estimatingthe Doppler spread (B⁽²⁾) by evaluating the Doppler spread function byusing the demodulated pilot symbol autocorrelation coefficients(r_(xx)(n).